small change

Manuscript/GW-srDFT.tex

@@ -235,7 +235,7 @@ $(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\tex
 \begin{equation}
 \bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
 \end{equation}
-with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
+with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
 
 %From Julien:
 %\begin{equation}
@@ -265,6 +265,14 @@ with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$
 
 \subsection{The GW Approximation}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+The Dyson equation can be written with an arbitrary reference
+\begin{equation}
+(G^{\cal B})^{-1} = (G_\text{ref}^{\cal B})^{-1}- \left( \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \Sigma_\text{ref}^{\cal B} \right)  - \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
+\end{equation}
+where $(G_\text{ref}^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1} - \Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = v_\text{Hxc}^{\cal B}(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
+
+
 Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
 More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.